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Hutchinson – Lai’s conjecture for bivariate extreme value copulas. (English) Zbl 1101.62340

Summary: The class of bivariate extreme value copulas, which satisfies the monotone regression positive dependence property or equivalently the stochastic increasing property, is considered. A variational calculus proof of the T. P. Hutchinson and C. D. Lai conjecture [Continuous bivariate distributions. Emphasizing applications. Adelaide (1990)] about Kendall’s tau and Spearman’s rho for this class is provided.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G32 Statistics of extreme values; tail inference
62H20 Measures of association (correlation, canonical correlation, etc.)
49N99 Miscellaneous topics in calculus of variations and optimal control
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[1] Benes, V., Stepen, J. (Eds.), 1997. Distributions with Given Marginals and Moment Problems. Kluwer Academic Publishers, Dordrecht.; Benes, V., Stepen, J. (Eds.), 1997. Distributions with Given Marginals and Moment Problems. Kluwer Academic Publishers, Dordrecht.
[2] Dacorogna, B., Direct Methods in the Calculus of Variations (1989), Springer: Springer Berlin · Zbl 0703.49001
[3] Dacorogna, B., 1992. Introduction au calcul des variations. Cahiers mathématiques de l’école polytechnique fédérale de Lausanne. Presses Polytechniques et Universitaires Romandes, Lausanne.; Dacorogna, B., 1992. Introduction au calcul des variations. Cahiers mathématiques de l’école polytechnique fédérale de Lausanne. Presses Polytechniques et Universitaires Romandes, Lausanne.
[4] Dall’Aglio, G., Kotz, S., Salinetti, G. (Eds.), 1991. Advances in Probability Distributions with Given Marginals. Kluwer Academic Publishers, Dordrecht.; Dall’Aglio, G., Kotz, S., Salinetti, G. (Eds.), 1991. Advances in Probability Distributions with Given Marginals. Kluwer Academic Publishers, Dordrecht.
[5] Daniels, H. E., Rank correlation and population models, J. Roy. Statist. Soc. Ser. B, 12, 171-181 (1950) · Zbl 0040.22302
[6] de Haan, L., A spectral representation for max-stable processes, Ann. Probab., 12, 1194-1204 (1984) · Zbl 0597.60050
[7] de Haan, L.; Resnick, S. I., Limit theory for multivariate sample extremes, Z. Wahrsch. Verw. Gebiete, 40, 317-337 (1977) · Zbl 0375.60031
[8] Deheuvels, P., 1978. Caractérisation compléte des lois extrêmes multivariées et de la convergence des types extrêmes. Publ. Inst. Statist. Univ. Paris 23 (3), 1-36.; Deheuvels, P., 1978. Caractérisation compléte des lois extrêmes multivariées et de la convergence des types extrêmes. Publ. Inst. Statist. Univ. Paris 23 (3), 1-36. · Zbl 0414.60043
[9] Deheuvels, P., Point processes and multivariate sample extreme values, J. Multivariate Anal., 13, 257-271 (1983) · Zbl 0519.60045
[10] Durbin, J.; Stuart, A., Inversions and rank correlation coefficients, J. Roy. Statist. Soc. Ser. B, 12, 303-309 (1951) · Zbl 0045.08405
[11] Embrechts, P.; Klüppelberg, C.; Mikosch, Th., Modelling Extremal Events (1997), Springer: Springer Berlin · Zbl 0873.62116
[12] Fisher, R. A.; Tippett, L. H.C., Limiting forms of the frequency distribution of the largest or smallest member of a sample, Proc. Camb. Phil. Soc., 24, 180-190 (1928) · JFM 54.0560.05
[13] Galambos, J., The Asymptotic Theory of Extreme Order Statistics (1987), Kreir: Kreir Melbourne, FL · Zbl 0634.62044
[14] Garralda Guillem, A.I., 2000. Structure de dépendance des lois de valeurs extrêmes bivariées. Comptes Rendus Acad. Sci. Paris 330, Série I, 593-596.; Garralda Guillem, A.I., 2000. Structure de dépendance des lois de valeurs extrêmes bivariées. Comptes Rendus Acad. Sci. Paris 330, Série I, 593-596. · Zbl 0951.60014
[15] Geoffroy, J., 1958/59. Contributions à la théorie des valeurs extrêmes. Publ. Inst. Statist. Univ. Paris 7/8, 37-185.; Geoffroy, J., 1958/59. Contributions à la théorie des valeurs extrêmes. Publ. Inst. Statist. Univ. Paris 7/8, 37-185.
[16] Ghoudi, K.; Khoudraji, A.; Rivest, L.-P., Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles, Canad. J. Statist., 26, 1, 187-197 (1998) · Zbl 0899.62071
[17] Hutchinson, T. P.; Lai, C. D., Continuous Bivariate Distributions, Emphasizing Applications (1990), Rumsby Scientific: Rumsby Scientific Adelaide · Zbl 1170.62330
[18] Joe, H., Multivariate Models and Dependence Concepts (1997), Chapmann & Hall: Chapmann & Hall London · Zbl 0990.62517
[19] Kimeldorf, G.; Sampson, A., One-parameter families of bivariate distributions with fixed marginals, Commun. Statist., 4, 293-301 (1975) · Zbl 0296.62012
[20] Kimeldorf, G.; Sampson, A., Positive dependence orderings, Ann. Inst. Statist. Math., 39, 113-128 (1987) · Zbl 0617.62006
[21] Konijn, H. S., Positive and negative dependence of two random variables, Sankhya, 21, 269-280 (1959) · Zbl 0089.15504
[22] Kruskal, W. H., Ordinal measures of association, J. Amer. Statist. Assoc., 53, 814-861 (1958) · Zbl 0087.15403
[23] Lehmann, E. L., Some concepts of dependence, Ann. Math. Statist., 37, 1137-1153 (1966) · Zbl 0146.40601
[24] Marshall, A. W.; Olkin, I., Domains of attraction of multivariate extreme value distributions, Ann. Probab., 11, 168-177 (1983) · Zbl 0508.60022
[25] Nelsen, R.B., 1991. Copulas and association. In: Dall’Aglio, G., Kotz, S., and Salinelli, G. (Eds.), Advances in Probability Distributions with Given Marginals. Kluwer Academic Publishers, Dordrecht, pp. 51-74.; Nelsen, R.B., 1991. Copulas and association. In: Dall’Aglio, G., Kotz, S., and Salinelli, G. (Eds.), Advances in Probability Distributions with Given Marginals. Kluwer Academic Publishers, Dordrecht, pp. 51-74. · Zbl 0727.60002
[26] Nelsen, R. B., An Introduction to Copulas (1999), Springer: Springer New York · Zbl 0909.62052
[27] Pickands, J., 1981. Multivariate extreme value distributions. Proceedings of the 43rd Session International Statistical Institute, Buenos Aires, pp. 859-878.; Pickands, J., 1981. Multivariate extreme value distributions. Proceedings of the 43rd Session International Statistical Institute, Buenos Aires, pp. 859-878. · Zbl 0518.62045
[28] Resnick, S. I., Extreme Values, Regular Variation, and Point Processes (1987), Springer: Springer Berlin · Zbl 0633.60001
[29] Rüschendorf, L., Schweizer, B., Taylor, M.D. (Eds.), 1996. Distributions with Fixed Marginals and Related Topics. Institute of Mathematical Statistics, Hayward.; Rüschendorf, L., Schweizer, B., Taylor, M.D. (Eds.), 1996. Distributions with Fixed Marginals and Related Topics. Institute of Mathematical Statistics, Hayward.
[30] Scarsini, M., On measures of condordance, Stochastics, 8, 201-218 (1984) · Zbl 0582.62047
[31] Sibuya, M., Bivariate extreme statistics, Ann. Inst. Statist. Math., 11, 195-210 (1960) · Zbl 0095.33703
[32] Sklar, A., Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8, 229-231 (1959) · Zbl 0100.14202
[33] Struwe, M., Variational Methods (1990), Springer: Springer Berlin
[34] Tawn, J. A., Bivariate extreme value theorymodels and estimation, Biometrika, 75, 397-415 (1988) · Zbl 0653.62045
[35] Tiago de Oliveira, J., 1958. Extremal distributions. Rev. Fac. Ciências Lisboa 2 Ser. A Mat. 7, 215-227.; Tiago de Oliveira, J., 1958. Extremal distributions. Rev. Fac. Ciências Lisboa 2 Ser. A Mat. 7, 215-227.
[36] Tiago de Oliveira, J., Bivariate models for extremes; statistical decision, (Tiago de Oliveira, J., Statistical Extremes and Applications (1984), Reidel: Reidel Dordrecht), 131-153 · Zbl 0574.62050
[37] Troutman, J. L., Variational calculus with elementary convexity (1983), Springer: Springer New York · Zbl 0523.49001
[38] Walter, W., 1972. Gewöhnliche Differentialgleichungen. Heidelberger Taschenbücher, Band 110. Springer, Berlin.; Walter, W., 1972. Gewöhnliche Differentialgleichungen. Heidelberger Taschenbücher, Band 110. Springer, Berlin.
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